Harmonic mean

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In mathematics, the harmonic mean (formerly sometimes called the subcontrary mean) is one of several kinds of average. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean H of the positive real numbers x₁, x₂, ..., x_n is defined to be

$H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^n 1/x_i}, \qquad x_i > 0 \text{ for all } i.$

Equivalently, the harmonic mean is the reciprocal of the arithmetic mean of the reciprocals.

[edit] Relationship with other means

A geometric construction of the three Pythagorean means (of two numbers only). Harmonic mean denoted by H in purple color.

The harmonic mean is one of the three Pythagorean means. For all data sets containing at least one pair of nonequal values, the harmonic mean is always the least of the three means, while the arithmetic mean is always the greatest of the three and the geometric mean is always in between. (If all values in a nonempty dataset are equal, the three means are always equal to one another; e.g. the harmonic, geometric, and arithmetic means of {2, 2, 2} are all 2.)

It is the special case M₋₁ of the power mean.

Since the harmonic mean of a list of numbers tends strongly toward the least elements of the list, it tends (compared to the arithmetic mean) to mitigate the impact of large outliers and aggravate the impact of small ones.

The arithmetic mean is often incorrectly used in places calling for the harmonic mean.^[1] In the speed example below for instance the arithmetic mean 50 is incorrect, and too big.

[edit] Weighted harmonic mean

If a set of weights $w 1$ , ..., $w n$ is associated to the dataset $x 1$ , ..., $x n$ , the weighted harmonic mean is defined by

$\sum_{i=1}^n w_i \bigg/ \sum_{i=1}^n \frac{w_i}{x_i}.$

The harmonic mean is the special case where all weights are equal to 1.

[edit] Examples

[edit] In physics

In certain situations, especially many situations involving rates and ratios, the harmonic mean provides the truest average. For instance, if a vehicle travels a certain distance at a speed x (e.g. 60 kilometres per hour) and then the same distance again at a speed y (e.g. 40 kilometres per hour), then its average speed is the harmonic mean of x and y (48 kilometres per hour), and its total travel time is the same as if it had traveled the whole distance at that average speed. However, if the vehicle travels for a certain amount of time at a speed x and then the same amount of time at a speed y, then its average speed is the arithmetic mean of x and y, which in the above example is 50 kilometres per hour. The same principle applies to more than two segments: given a series of sub-trips at different speeds, if each sub-trip covers the same distance, then the average speed is the harmonic mean of all the sub-trip speeds, and if each sub-trip takes the same amount of time, then the average speed is the arithmetic mean of all the sub-trip speeds. (If neither is the case, then a weighted harmonic mean or weighted arithmetic mean is needed.)

Similarly, if one connects two electrical resistors in parallel, one having resistance x (e.g. 60Ω) and one having resistance y (e.g. 40Ω), then the effect is the same as if one had used two resistors with the same resistance, both equal to the harmonic mean of x and y (48Ω): the equivalent resistance in either case is 24Ω (one-half of the harmonic mean). However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (with total resistance equal to the sum of x and y). And, as with previous example, the same principle applies when more than two resistors are connected, provided that all are in parallel or all are in series.

[edit] In other sciences

In Information retrieval and some other fields, the harmonic mean of the precision and the recall is often used as an aggregated performance score: the F-score (or F-measure).

An interesting consequence arises from basic algebra in problems of working together. As an example, if a gas-powered pump can drain a pool in 4 hours and a battery-powered pump can drain the same pool in 6 hours, then it will take both pumps 6 · 4/(6 + 4), which is equal to 2.4 hours, to drain the pool together. Interestingly, this is one-half of the harmonic mean of 6 and 4.

In hydrology the harmonic mean is used to average hydraulic conductivity values for flow that is perpendicular to layers (e.g. geologic or soil). On the other hand, for flow parallel to layers the arithmetic mean is used.